5-3 Polynomial Functions
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1 For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree function, and c. state the number of real zeros. 35. a. As the x-values approach negative infinity, the y-values approach positive infinity. As the x-values approach positive infinity, the y-values approach positive infinity. The end behavior is in the same direction. b. Since the end behavior is in same direction, it is an even-degree function. c. The graph intersects the x-axis at four points, so there are four real zeros. 36. a. As the x-values approach negative infinity, the y-values approach positive infinity. As the x-values approach positive infinity, the y-values approach negative infinity. The end behavior is in opposite direction. b. Since the end behavior is in opposite direction, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. esolutions Manual - Powered by Cognero Page 1
2 37. a. As the x-values approach negative infinity, the y-values approach negative infinity. As the x-values approach positive infinity, the y-values approach positive infinity. The end behavior is in opposite direction. b. Since the end behavior is in opposite direction, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. esolutions Manual - Powered by Cognero Page 2
3 Use the degree and end behavior to match each polynomial to its graph. A. B. C. D. 47. f(x) = x 3 + 3x 2 4x Since the degree of the polynomial is 3, the graph must have 2 turning points. And the leading coefficient of the graph is positive. Therefore, the correct choice is D. 48. f (x) = 2x 2 + 8x + 5 Since the degree of the polynomial is 2, the graph must have 1 turning point. And the function is an even degree function. Therefore, the end behavior of the graph must be in the same direction. The correct choice is B. esolutions Manual - Powered by Cognero Page 3
4 49. f (x) = x 4 3x 2 + 6x Since the degree of the polynomial is 4, the graph must have 3 turning points. And the function is an even degree function. Therefore, the end behavior of the graph must be in the same direction. The correct choice is A. 50. f (x) = 4x 3 4x Since the degree of the polynomial is 3, the graph must have 2 turning point2. And the leading coefficient of the graph is negative. Therefore, the graph opens down. The correct choice is B. 63. CCSS CRITIQUE Shenequa and Virginia are determining the number of real zeros of the graph. Is either of them correct? Explain your reasoning. A double zero still represents 1 real zero. You will learn in a later lesson that a double zero is an indication of a multiple linear factor. Since the graph intersects the x-axis 7 times, the polynomial will have exactly 7 real zeros. Therefore, Shenequa is correct. esolutions Manual - Powered by Cognero Page 4
5 64. CHALLENGE Of f (x) and g(x), which function has more potential real zeros? What is the degree of that function? g(x) = x 4 + x 3 13x 2 + x + 4 The sign of f (x) changes for 5 times. So, f (x) has the potential for 5 or more real zeros and a degree of 5 or more. Since g(x) has a degree of 4, it has the potential for 4 real zeros. So, f (x) has more potential real zeros. 65. CHALLENGE If f (x) has a degree of 5 and a positive leading coefficient and g(x) has a degree of 3 and a positive leading coefficient, determine the end behavior of.explain your reasoning. Sample answer: Since f (x) has a degree of 5 and g(x) has a degree of 3, then will have a degree of 2. Since both functions have positive leading coefficients, their quotient will also have a positive leading coefficient. Therefore, the end behavior is: will become a 2nd-degree function with a positive leading coefficient. esolutions Manual - Powered by Cognero Page 5
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3.2 Graphs of Higher Degree Polynomial Functions Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n-1,,a 2, a 1, a 0, be real numbers with a n 0. The function defined by